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Albert Gatt LIN3021 Formal Semantics Lecture 1. What this study unit is about We are concerned (once again) with the question of meaning. Our primary.

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Presentation on theme: "Albert Gatt LIN3021 Formal Semantics Lecture 1. What this study unit is about We are concerned (once again) with the question of meaning. Our primary."— Presentation transcript:

1 Albert Gatt LIN3021 Formal Semantics Lecture 1

2 What this study unit is about We are concerned (once again) with the question of meaning. Our primary aim will be to look in depth at a number of selected topics, especially: Predication Reference Quantification Tense, aspect and modality

3 Why “formal”? The sentence snow is white is true iff snow is white A sentence in the object language (English in this case) A sentence about the sentence in the object language.

4 Why “formal” The sentence snow is white is true iff snow is white The above sentence makes a statement about a sentence. It’s a meta-sentence. (The meta-sentence is in English, but that’s purely incidental.) When we talk about meaning, we need a meta-language. We could use English, but this can lead to ambiguity. We’d like to be more precise. A formal meta-language is a language that has well-understood structural and semantic properties. A lot of what we’ll do will be couched in the language of logic.

5 Richard Montague Logician whose work in the 1970s is largely credited with having given birth to the enterprise of formal semantics. I reject the contention that an important theoretical difference exists between formal and natural languages. (Montague 1970) His main contribution was in making use of the formal tools of logic and mathematics to analyse meaning. This approach was explicitly non-psychological (Montague was in agreement with Frege here). So there’s a strong contrast with the psychological orientation of linguistic theory, especially after Chomsky. That hasn’t prevented linguists and other formal semanticists from collaborating.

6 What is a “meaning”? Whether we’re looking at sentences, or smaller linguistic units (words, morphemes, phrases...), we need first to decide what it is we mean (!) when we talk about “meaning”. We’ll first consider three possible definitions: 1. The meaning of expression X is just its relationship with other expressions (Y, Z,...) 2. The meaning of expression X is something in the mind/brain (a concept, an idea). 3. The meaning of expression X is just its “use”: that complex of social habits whereby we use X in a certain way.

7 Theory 1 Meaning holism

8 The classic statement: Our statements about the external world face the tribunal of sense experience not individually but only as a corporate body. (W.V.O. Quine, 1953) We can think of language (and other linguistic practices, including science) as a dense network of interconnected meanings. The meaning of an expression is determined on the basis of its position within this network.

9 A definition Meaning holism is the doctrine that the meaning of a linguistic expression (word, sentence) is not determined in isolation, but in relation to many other expressions (words, sentences). To know what man means, you must know what human means and what biped means and... To understand the statement Force is mass times velocity you need to understand a whole lot of other statements. (More on this later).

10 A motivating example Force is mass times velocity This is a scientific statement. It is not usually learned on its own. It only makes sense within a theory (= an entire body of related statements). The term force can only be understood if we know the place of the terms mass and velocity within the theory. But even mass and velocity are understood relative to the rest of the theory Similarly, words like tall tend to be understood (or at least explained) in relation to words like short.

11 Problems If meaning is defined in terms of “relations to other linguistic units”, then we run into some problems: Can we have a scientific semantics? Which relations count and which do not? Do we really want to admit arbitrary meaning relations into our theory?

12 Theory 2 Meaning as social habit (or game)

13 The classic statement For a large class of cases — though not for all — in which we employ the word ‘meaning’ it can be defined thus: the meaning of a word is its use in the language (Wittgenstein, 1953: par. 43).

14 An example What does “It’s raining” mean? Here are some ways in which we use it: Statement of fact (I hereby inform you that it’s raining) Query/request (Please give me an umbrella, shut the window...) Counter-proposal (I don’t want to go out).... Our knowledge of the meaning of the sentence is our knowledge of (a) how it is used; (b) what to do in response to it, in the various situations in which it’s used. A bit like a game, where a move motivates a counter-move...

15 Problems This theory suggests that languages are (sets of) arbitrary conventions. Contemporary linguistic theory has suggested that there are (at least some) universals that constrain the types of languages humans use. Under this theory, it’s difficult to see why some responses to some messages are appropriate, while others are not. Why is It’s 10pm appropriate as a reply for What’s the time? Why don’t we arbitrarily assume a convention that motivates the reply I like sugar? Why can’t we change the rules of the game?

16 Theory 3 Meaning as concept/idea

17 A characterisation It seems intuitively obvious that something like this relation holds between linguistic expressions, our minds and the world. CONCEPT (sense) expression means objects determines denotes

18 The question We won’t dispute the fact that we have some mental representation of the conceptual content of expressions. The question is: is this what we mean by “meaning” (i.e. Is this useful for a scientific semantics)? mental representation things & situations linguistic expressions

19 The Twin Earth chronicles (Putnam, 1975) Earth and Twin Earth: 100% identical in all respects (including that there’s someone there now following a lecture on formal semantics, who is named, looks and thinks exactly like you) Except that: Water on Earth = H2O Water on Twin Earth = XYZ (Otherwise, they look, taste and feel exactly the same) When you and your Twin Earth counterpart say water, you are able to identify the substance on your respective planets. The substances look, feel etc exactly the same, but they are not. But you and your counterpart are identical in all respects (even mentally)! Therefore, while you mean something different in reality, this is not because you have different concepts. Therefore, the meaning of water can’t be mental.

20 Putnam’s conclusion “Cut the pie any way you like, meaning just ain’t in the head!” (Putnam, 1975) The point (for us) of this thought experiment is that thinking of meaning in purely mental terms might not be that useful if we want an objective theory. So what is the crucial component of meaning? When we use an expression (e.g. dog), we intend to mean certain things in the world. So the crucial thing might be, the things in the world that we mean.

21 The upshot We’re not arguing against the existence of concepts/mental constructs. We’re simply saying that, from a semantic p.o.v, we might as well ignore them. mental representation things & situations linguistic expressions

22 An alternative view Meaning is “out there”

23 Let’s take a closer look Suppose we say that: The meaning of dog is whatever it is that describes all the things that are really dogs, out there in the world. Notice that we’re thinking of meaning as independent of our own knowledge. (It doesn’t matter if we don’t know how to formulate the meaning itself – it exists independently of us) Perhaps this might work fine for words (especially nouns), but what about sentences (It’s raining), or properties (blue)?

24 Sentences, worlds and truth conditions The boy kissed the girl. As an English speaker, you know that this sentence is true in situation (“world”) A, but not in B. AB

25 More generally... TrueFalse The boy kissed the girl.

26 Truth conditions What these examples show is that knowing the meaning of a sentence involves (at least) knowing the conditions under which that sentence is true. Just like knowing the meaning of dog involves knowing what things in the world are dogs... NB: we are not saying that knowing the meaning of a sentence means knowing whether it’s true or false!

27 Possible worlds We can think of each of these scenarios as a “world” More accurately, they depict only one situation of interest in each possible world. Each of these worlds is much more complex. There’s an infinity of possible worlds. (Try and think why this could be the case.)

28 A sentence “means” those worlds in which it’s true Recall that, ultimately, when we talk about sentence meaning, we’re interested in propositions A proposition can be equated with the set of worlds in which that proposition is true.

29 In graphics... In other words, a sentence (proposition) describes a set of worlds, those in which it is true. (Again compare to a noun like dog, which denotes the set of things of which the noun is “true”, i.e. the set of things which are dogs). Proposition means Worlds in which the proposition is true

30 Three arguments for possible worlds If we think of sentence meaning in this way, then we can get a handle on the meaning of “logical” words like and, or, not 2....we can give a precise account of sentential relationships like synonymy and entailment 3....we can also deal with some aspects of human action and agency in a rational way

31 The meaning of logical connectives Think of linguistic “and” as something like logical conjunction. Intuitively, the meaning of p  q is the intersection of the set of worlds in which p is true and q is true. [The circle is inside the square] and [the circle is yellow]. Worlds where p is true Worlds where q is true

32 So what about......the meaning of: pVq (disjunction) ¬p (negation) Can we always assume that conjunction works like this? Jack kissed Mary and Susan

33 Semantic relations between propositions I: synonymy P and q are synonymous if they’re true in exactly the same set of worlds: P: the circle is larger than the square Q: the square is smaller than the circle It’s easy to see that these propositions are true in exactly the same worlds (and false in exactly the same worlds too) W w1 w2 w3

34 Semantic relations between propositions I: entailment P entails q if q is true whenever p is true: P: the circle is inside the square Q: the circle is smaller than the square W

35 What about......when p and q are contrary (i.e. Can’t both be true)?...when p and q are contradictory (i.e. Can’t both be true and can’t both be false)

36 Meaning and human action What’s the primary purpose of language? We might think that the main purpose of linguistic communication is to pass on information. There are other uses, of course (including more “playful” uses). The reception of information impacts our beliefs about the world: We can reformulate (or even discard) beliefs We can form new ones We can confirm old ones Human action is founded on (rational) belief.

37 An example Suppose that: Mary believes it’s going to be sunny today. Mary wishes to go out. Mary knows that if it’s sunny, and she goes out, she doesn’t need an umbrella. But she does if it’s rainy. Lindsey says: The weather report said it’s going to rain. If Mary believes Lindsey, then she will have to update her beliefs about the world. This will also impact the course of action she chooses, based on her desires.

38 Example cont/d Before LindseyAfter Lindsey Mary’s belief worldsMary’s desire worlds Mary’s belief worldsMary’s desire worlds

39 Beyond declaratives Can we extend the possible worlds framework to deal with other types of sentences? Questions: What did you have for breakfast? Imperatives: Eat your breakfast!...

40 Questions Observe that questions are requests for information. The form of the question hints at what sort of information ought to be given. Suppose we think of a question as meaning the set of propositions that are possible answers to the question. What did you have for breakfast? Proposition p: I had bacon and eggs Proposition q: I had cereal Proposition r: I had bacon and eggs and cereal

41 Orders Recall that as a sentence allows us to categorise worlds into those where the sentence is true and those where it’s false. We could think of imperatives as categorising worlds into those where the order is carried out and those where it isn’t. Eat your breakfast! Worlds in which I eat my breakfast Worlds in which I don’t eat my breakfast

42 From worlds to models (a reminder of some things covered in LIN1032)

43 Propositions as functions What we’ve said so far about propositions could be formalised as follows: A proposition is a function from possible worlds to truth values. In other words, understanding a proposition means being able to check, for any conceivable possible world, whether the proposition is true in that world or not.

44 Propositions as functions (II) This situation could be modelled like this: In other words, we can view a proposition as a function from possible worlds to truth values. Proposition T F

45 Model theory The kind of semantics we’ll be doing is often called model- theoretic. That’s because we assume that interpretation is carried out relative to a model of the world. A model is: A structured domain of the relevant entities which allows us to interpret all the expressions of our meta-language. For now, think of it as a “small partial model of the world” A natural language expression will be translated into the (logical) meta-language and interpreted according to the model.

46 Models Components of a model: a universe of individuals U an interpretation function I which assigns semantic values to our constants. the truth values {T,F} as usual (for propositions) Formally: M = “Model M is made up of U and I”

47 Constructing a world Suppose our example world contains exactly 4 individuals. U = {Isabel Osmond, Emma Bovary, Alexander Portnoy, Beowulf}

48 Assigning referents to constants To each individual constant, there corresponds some individual in the world, as determined by the interpretation function I: [[a]] M = Isabel Osmond [[b]] M = Emma Bovary [[c]] M = Alexander Portnoy [[d]] M = Beowulf

49 Predicates We also have a fixed set of predicates in our meta-language. These correspond to natural language expressions. Their interpretation (extension) needs to be fixed for our model too: 1-place predicates are sets of individuals [[P]] M  U 2-place predicates are sets of ordered pairs [[Q]] M  U x U 3-place predicates are sets of triples: [[R]] M  U x U x U … and so on

50 Assigning extensions to predicates Suppose we have 2 1-place predicates in our language: tall and clever Let us fix their extension like this: [[tall]] M = {Emma Bovary, Beowulf} [[clever]] M = {Alexander Portnoy, Emma Bovary, Isabel Osmond} Thus, in our world, we know the truth of the following propositions: [[tall(a)]] M = FALSE (since Isabel isn’t tall) [[clever(c)]] M = TRUE (since Alexander is clever)

51 More formally… A sentence of the form P(t), where P is a predicate and t is an individual term (constant or variable) is true in some model M = iff: the object assigned to t by I is in the extension of P (i.e. the object that t points to is in the set of things of which P is true) i.e. t  [[P]] M How would you extend this to a sentence of the form P(t1, t2), using a 2-place predicate?

52 For two-place predicates A sentence of the form P(t1, t2), where P is a predicate and t1, t2 are individual terms (constants or variables) is true in some model M iff: the ordered pair of the objects assigned to t1 and t2 can be found among the set of ordered pairs assigned to P in that interpretation

53 Propositions We evaluate propositions against our model, and determine whether they’re true or false. If β is a proposition, then [[ β ]] M is either the value TRUE or the value FALSE, i.e.: [[ β ]] M  {TRUE, FALSE}

54 Complex formulas Once we have constructed our interpretation (assigning extensions to predicates and values to constants), complex formulas involving connectives can easily be interpreted. We just compute the truth of a proposition based on the connectives and the truth of the components. Recall that connectives have truth tables associated with them. For propositions that contain connectives, the truth tables describe the function from worlds to truth values.

55 Truth tables and models For a complex proposition (e.g. p  q), our truth table tells us in which worlds this proposition would be true or false, namely: p  q is true in any world where p is false and q is false, or p is true and q is true, etc

56 Complex formulas: example Model: [[a]] M = I. Osmond [[b]] M = E. Bovary [[c]] M = A. Portnoy [[d]] M = Beowulf [[tall]] M = {E. Bovary, Beowulf} [[clever]] M = {A. Portnoy, E. Bovary, I. Osmond} Formulas: [[tall(a) Λ clever(b)]] M “Isabel is tall and Emma is clever” FALSE [[clever(a) ν tall(a)]] M Isabel is tall or clever TRUE [[¬tall(a)]] M Isabel is not tall TRUE

57 Summary For the remainder of this course, we’re going to (tentatively) assume that: Meaning exists independently of minds and individuals Possible worlds present a good formalism for dealing with meanings and meaning relationships. Next week, we’ll discuss the concept of compositionality in some detail and we’ll also continue to flesh out our model- theoretic conception of meaning.


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